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Intermediate Algebra: Functions and Graphs

Katherine Yoshiwara

Section 9.6 Chapter 9 Summary and Review

Subsection 9.6.1 Glossary

  • horizontal
  • vertical
  • parallel
  • perpendicular
  • conic section
  • circle
  • ellipse
  • parabola
  • hyperbola
  • central conic
  • major axis
  • minor axis
  • vertices
  • covertices
  • transverse axis
  • conjugate axis
  • asymptote
  • branch

Subsection 9.6.2 Key Concepts

  1. Horizontal and Vertical Lines.

    1. The equation of the horizontal line passing through \((0, b)\) is
      \begin{gather*} y=b \end{gather*}
    2. The equation of the vertical line passing through \((a,0)\) is
      \begin{gather*} x=a \end{gather*}

  2. Slopes of Horizontal and Vertical Lines.

    The slope of a horizontal line is zero.
    The slope of a vertical line is undefined.

  3. Parallel and Perpendicular Lines.

    1. Two lines are parallel if their slopes are equal, that is, if
      \begin{gather*} m_1 = m_2 \end{gather*}
      or if both lines are vertical.
    2. Two lines are perpendicular if the product of their slopes is \(-1\text{,}\) that is, if
      \begin{gather*} m_1m_2 = -1 \end{gather*}
      or if one of the lines is horizontal and one is vertical.
  4. The distance between two points is the length of the segment joining them.

  5. Distance Formula.

    The distance \(d\) between points \(P_1(x_1, y_1)\) and \(P_2(x_2, y_2)\) is
    \begin{gather*} d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \end{gather*}
  6. The midpoint of a segment is the point halfway between its endpoints, so that the distance from the midpoint to either endpoint is the same.

  7. Midpoint Formula.

    The midpoint of the line segment joining the points \(P_1(x_1, y_1)\) and \(P_2(x_2, y_2)\) is the point \(M(\overline{x},\overline{y})\) where
    \begin{gather*} \overline{x}=\frac{x_1+x_2}{2} \quad\text{ and } \overline{y}=\frac{y_1+y_2}{2} \end{gather*}
  8. A circle is the set of all points in a plane that lie at a given distance, called the radius, from a fixed point called the center.

  9. Circle.

    The equation for a circle of radius \(r\) centered at the point \((h, k)\) is
    \begin{gather*} (x-h)^2+(y-k)^2=r^2 \end{gather*}

  10. Ellipse.

    The standard form for an ellipse centered at \((h, k)\) is
    \begin{gather*} \frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1 \end{gather*}

  11. Hyperbola.

    The equation for a hyperbola centered at the point \((h, k)\) has one of the two standard forms:
    \begin{gather*} \frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1 \end{gather*}
    \begin{gather*} \frac{(y-k)^2}{b^2}-\frac{(x-h)^2}{a^2}=1 \end{gather*}

  12. Conic Sections.

    The graph of \(Ax^2+Cy^2+Dx+Ey+F=0\) is
    1. a circle of \(A=C\text{.}\)
    2. a parabola if \(A=0\) or \(C=0\) (but not both).
    3. an ellipse if \(A\) and \(C\) have the same sign.
    4. a hyperbola if \(A\) and \(C\) have opposite signs.
  13. A system of two quadratic equations may have up to four solutions.

Exercises 9.6.3 Chapter 9 Review Problems

Exercise Group.

For Problems 1 and 2, decide whether the lines are parallel, perpendicular, or neither.
1.
\(y=\dfrac{1}{2}x+3 \text{;}\) \(\quad x-2y=8\)
Answer.
parallel
2.
\(4x-y=6 \text{;}\) \(\quad x+4y=-2\)
Answer.
perpendicular

3.

Write an equation for the line that is parallel to the graph of \(2x + 3y = 6\) and passes through the point \((1, 4)\)
Answer.
\(y=\dfrac{-2}{3}x+\dfrac{14}{3} \)

4.

Write an equation for the line that is perpendicular to the graph of \(2x + 3y = 6\) and passes through the point \((1, 4)\)
Answer.
\(y=\dfrac{3}{2}x+\dfrac{5}{2} \)

5.

Two vertices of the rectangle \(ABCD\) are \(A(3, 2)\) and \(B(7, −4)\text{.}\) Find an equation for the line that includes side \(\overline{BC}\text{.}\)
Answer.
\(y=\dfrac{2}{3}x-\dfrac{26}{3} \)

6.

One leg of the right triangle \(PQR\) has vertices \(P(-8, -1)\) and \(Q(-2, -5)\text{.}\) Find an equation for the line that includes the leg \(\overline{QR} \text{.}\)
Answer.
\(y=\dfrac{3}{2}x-2 \)

7.

Find the perimeter of the triangle with vertices \(A(-1, 2)\text{,}\) \(B(5, 4)\text{,}\) \(C(1, -4)\text{.}\) Is \(\Delta ABC\) a right triangle?
Answer.
21.59; yes

8.

Find the midpoint of each side of \(\Delta ABC\) from the previous problem. Join the midpoints to form a new triangle, and find its perimeter.
Answer.
10.8

Exercise Group.

For Problems 9–18, graph the conic section.
9.
\(x^2+y^2=9\)
Answer.
circle
10.
\(\dfrac{x^2}{9}+y^2=1\)
Answer.
ellipse
11.
\(\dfrac{x^2}{4}+\dfrac{y^2}{9}=1\)
Answer.
ellipse
12.
\(\dfrac{y^2}{6}-\dfrac{x^2}{8}=1\)
Answer.
hyperbola
13.
\(4(y-2)=(x+3)^2\)
Answer.
parabola
14.
\((x-2)^2+(y+3)^2=16\)
Answer.
circle
15.
\(\dfrac{(x-2)^2}{4}+\dfrac{(y+3)^2}{9}=1\)
Answer.
ellipse
16.
\(\dfrac{(x+4)^2}{12}+\dfrac{(y-2)^2}{6}=1\)
Answer.
ellipse
17.
\(\dfrac{(x-2)^2}{4}+\dfrac{(y+3)^2}{9}=1\)
Answer.
hyperbola
18.
\((x-2)^2+4y=4\)
Answer.
parabola

Exercise Group.

For Problems 19–30
  1. Write the equation of each conic section in standard form.
  2. Identify the conic and describe its main features.
19.
\(x^2+y^2-4x+2y-4=0\)
Answer.
  1. \(\displaystyle (x-2)^2+(y+1)^2=9\)
  2. Circle: center \((2,-1)\text{,}\) radius 3
20.
\(x^2+y^2-6y-4=0\)
Answer.
  1. \(\displaystyle x^2+(y-3)^2=13\)
  2. Circle: center \((0,3)\text{,}\) radius \(\sqrt{13}\)
21.
\(4x^2+y^2-16x+4y+4=0\)
Answer.
  1. \(\displaystyle \dfrac{(x-2)^2}{4}+\dfrac{(y+2)^2}{16}=1\)
  2. Ellipse: center \((2,-2)\text{,}\) \(a=2\text{,}\) \(b=4\)
22.
\(8x^2+5y^2+16x-20y-12=0\)
Answer.
  1. \(\displaystyle \dfrac{(x+1)^2}{5}+\dfrac{(y-2)^2}{8}=1\)
  2. Ellipse: center \((-1,2)\text{,}\) \(a=\sqrt{5}\text{,}\) \(b=\sqrt{8}\)
23.
\(x^2-8x-y+6=0\)
Answer.
  1. \(\displaystyle y+10=(x-4)^2\)
  2. Parabola: vertex \((4,10)\text{,}\) opens upward, \(a=1\)
24.
\(y^2+6y+4x+1=0\)
Answer.
  1. \(\displaystyle x-2=\dfrac{-1}{4}(y+3)^2\)
  2. Parabola: vertex \((2,-3)\text{,}\) opens left, \(a=\dfrac{-1}{4}\)
25.
\(x^2+y=4x-6\)
Answer.
  1. \(\displaystyle y+2=-(x-2)^2\)
  2. Parabola: vertex \((2,-2)\text{,}\) opens downward, \(a=-1\)
26.
\(y^2=2y+2x+2\)
Answer.
  1. \(\displaystyle x+\dfrac{3}{2}=\dfrac{1}{2}(y-1)^2\)
  2. Parabola: vertex \(\left(\dfrac{-3}{2},1\right)\text{,}\) opens right, \(a=\dfrac{1}{2}\)
27.
\(2y^2-3x^2-16y-12x+8=0\)
Answer.
  1. \(\displaystyle \dfrac{(y-4)^2}{6}-\dfrac{(x+2)^2}{4}=1\)
  2. Hyperbola: center \((-2,4)\text{,}\) transverse axis vertical, \(a=2\text{,}\) \(b=\sqrt{6}\)
28.
\(9x^2-4y^2-72x-24y+72=0\)
Answer.
  1. \(\displaystyle \dfrac{(x-4)^2}{4}-\dfrac{(y+3)^2}{9}=1\)
  2. Hyperbola: center \((4,-3)\text{,}\) transverse axis horizontal, \(a=2\text{,}\) \(b=3\)
29.
\(2x^2-y^2+6y-19=0\)
Answer.
  1. \(\displaystyle \dfrac{x^2}{5}-\dfrac{(y-3)^2}{10}=1\)
  2. Hyperbola: center \((0,3)\text{,}\) transverse axis horizontal, \(a=\sqrt{5}\text{,}\) \(b=\sqrt{10}\)
30.
\(4y^2-x^2+8x-28=0\)
Answer.
  1. \(\displaystyle \dfrac{y^2}{3}-\dfrac{(x-4)^2}{12}=1\)
  2. Hyperbola: center \((4,0)\text{,}\) transverse axis vertical, \(a=2\sqrt{3}\text{,}\) \(b=\sqrt{310}\)

31.

An ellipse centered at the origin has a vertical major axis of length 16 and a horizontal minor axis of length 10.
  1. Find the equation of the ellipse.
  2. What are the values of \(y\) when \(x = 4\text{?}\)
Answer.
  1. \(\displaystyle \dfrac{x^2}{25}+\dfrac{y^2}{64}=1\)
  2. \(\displaystyle \dfrac{\pm 24}{5}\)

32.

An ellipse centered at the origin has a horizontal major axis of length 26 and a vertical minor axis of length 18.
  1. Find the equation of the ellipse.
  2. What are the values of \(y\) when \(x = 12\text{?}\)
Answer.
  1. \(\displaystyle \dfrac{x^2}{169}+\dfrac{y^2}{81}=1\)
  2. \(\displaystyle \dfrac{\pm 45}{13}\)

Exercise Group.

For Problems 33–38, write an equation for the conic section with the given properties.
33.
Circle: center at \((-4, 3)\text{,}\) radius \(2\sqrt{5}\)
Answer.
\((x+4)^2+(y-3)^2=20 \)
34.
Circle: endpoints of a diameter at \((-5, 2)\) and \((1, 6)\)
Answer.
\((x+2)^2+(y-4)^2=13 \)
35.
Ellipse: center at \((-1, 4)\text{,}\) \(a = 4\text{,}\) \(b = 2\)
Answer.
\(\dfrac{(x+1)^2}{16}+\dfrac{(y-4)^2}{4}=1 \)
36.
Ellipse: vertices at \((3, 6)\) and \((3, -4)\text{,}\) covertices at \((1, 1)\) and \((5, 1)\)
Answer.
\(\dfrac{(x-3)^2}{4}+\dfrac{(y-1)^2}{25}=1 \)
37.
Hyperbola: center \((2, -3)\text{,}\) \(a = 4\text{,}\) \(b = 3\text{,}\) transverse axis horizontal
Answer.
\(\dfrac{(x-2)^2}{16}-\dfrac{(y+3)^2}{9}=1 \)
38.
Hyperbola: one vertex at \((−4, 1)\text{,}\) one end of the vertical conjugate axis at \((-3, 4)\)
Answer.
\((x+3)^2 -\dfrac{(y-1)^2}{9}=1 \)

Exercise Group.

For Problems 39–42, graph the system of equations and state the solutions of the system
39.
\(\begin{aligned}[t] 4x^2+y^2=25\\ x^2-y^2=-5 \end{aligned}\)
Answer.
\((\pm 2, \pm 3) \)
40.
\(\begin{aligned}[t] 4x^2-9y^2+132=0\\ x^2+4y^2-67=0 \end{aligned}\)
Answer.
\((\pm\sqrt{3}, \pm 4) \)
41.
\(\begin{aligned}[t] x^2+3y^2=13\\ xy=-2 \end{aligned}\)
Answer.
\((1,-2) \text{,}\) \((-1,2) \text{,}\) \(\left(2\sqrt{3}, \dfrac{-1}{\sqrt{3}} \right) \text{,}\) \(\left(-2\sqrt{3}, \dfrac{1}{\sqrt{3}} \right) \)
42.
\(\begin{aligned}[t] x^2+y^2=17\\ 2xy=-17 \end{aligned}\)
Answer.
\(\left(\dfrac{\sqrt{34}}{2}, \dfrac{\sqrt{34}}{2} \right) \text{,}\) \(\left(\dfrac{-\sqrt{34}}{2}, \dfrac{-\sqrt{34}}{2} \right) \)

Exercise Group.

For Problems 43–48, write and solve an equation or a system of equations
43.
Moia drives 180 miles in the same time that Fran drives 200 miles. Find the speed of each if Fran drives 5 miles per hour faster than Moia.
Answer.
Moia: 45 mph, Fran: 50 mph
44.
The perimeter of a rectangle is 26 inches and the area is 12 square inches. Find the dimensions of the rectangle.
Answer.
12 in by 1 in
45.
The perimeter of a rectangle is 34 centimeters and the area is 70 square centimeters. Find the dimensions of the rectangle.
Answer.
7 cm by 10 cm
46.
A rectangle has a perimeter of 18 feet. If the length is decreased by 5 feet and the width is increased by 12 feet, the area is doubled. Find the dimensions of the original rectangle.
Answer.
7 ft by 2 ft
47.
Norm takes a commuter train 10 miles to his job in the city. The evening train returns him home at a rate 10 miles per hour faster than the morning traintakes him to work. If Norm spends a total of 50 minutes per day commuting, what is the rate of each train?
Answer.
Morning train: 20 mph, evening train: 30 mph
48.
Hattie’s annual income from an investment is $32. If she had invested $200 more and the rate had been 1/2% less, her annual income would have been $35. What are the amount and rate of Hattie’s investment?
Answer.
Amount: $800, rate: 4%
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